Problem: All of the 3rd grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$4.00$ each for students, and the group paid $$53.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$19.50$ each for teachers and $$7.50$ each for students, and the group paid $$114.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+4y = 53}$ ${19.5x+7.5y = 114}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-19.5x-12y = -159}$ ${19.5x+7.5y = 114}$ Add the top and bottom equations together. $ -4.5y = -45 $ $ y = \dfrac{-45}{-4.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+4y = 53}$ to find $x$ ${6.5x + 4}{(10)}{= 53}$ $6.5x+40 = 53$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {19.5x+7.5y = 114}$ and get the same answer for $x$ ${19.5x + 7.5}{(10)}{= 114}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.